Magnus and Dyson Series for Master Integrals

We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 -> 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.Comment: 32 pages, 6 figures, 2 ancillary files. v2: references added, typos corrected in the text and in the ancillary file

Feynman Diagrams and Differential Equations

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D=4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.Comment: invited review article from Int. J. Mod. Phys.

First Principle Study of Capping Energies and Electronic States in Stoichiometric and Nonstoichiometric PbSe Nanoclusters

A large variety of PbSe nanoclusters have been modeled at the DFT level, to study their structure, their affinity for different ligands and their electronic properties, also depending on surface passivation. The clusters are extracted from the bulk rock salt structure with cubic, prism, truncated cubic, cuboctahedral and octahedral shape and they are fully relaxed, before computing the addition energies of methylamine and formate anions in different positions, to model the process of surface passivation. Then the density of states of all the clusters is computed, to study in particular the band gap and the behavior of the so-called intragap states, which affect the photophysical properties of the nanoparticles, also acting as trap states for charge carriers. We confirm the strong relationship between nanocluster off-stoichiometry and intragap states: such states can be localized on the surface, in the bulk or delocalized over the nanoparticle, according to the source of off-stoichiometry. The ability of different ligands to eliminate the intragap states are tested and discussed, also proposing nonstandard capping molecules

Elastic rods in life- and material-sciences: A general integrable model

The study of elastic deformations in thin rods has recently seen renewed interest due to the close connection between these systems and coarse-grained models of widespread application in life- and material-sciences. Until now, the analysis has been restricted to the solution of equilibrium equations for continuous models characterized by constant bending and twisting elastic moduli and/or by isotropic rod section. However, more realistic models often require more general conditions: indeed this is the case whenever microscopic information issuing from atomistic simulations is to be transferred to analytic or semi-analytic coarse-grained or macroscopic models. In this paper we will show that integrable, indeed solvable, equations are obtained under quite general conditions and that regular (e.g. circular helical) solutions emerge from reasonable choices of elastic stiffnesses

Elastic rods in life- and material-sciences: A general integrable model

The study of elastic deformations in thin rods has recently seen renewed interest due to the close connection between these systems and coarse-grained models of widespread application in life- and material-sciences. Until now, the analysis has been restricted to the solution of equilibrium equations for continuous models characterized by constant bending and twisting elastic moduli and/or by isotropic rod section. However, more realistic models often require more general conditions: indeed this is the case whenever microscopic information issuing from atomistic simulations is to be transferred to analytic or semi-analytic coarse-grained or macroscopic models. In this paper we will show that integrable, indeed solvable, equations are obtained under quite general conditions and that regular (e.g. circular helical) solutions emerge from reasonable choices of elastic stiffnesses

Existence of energy minimums for thin elastic rods in static helical configurations

We characterize families of solutions of the static Kirchhoff model of a thin elastic rod physically. These families, which are proved to exist, depend on the behavior of the so-called register and also on the radius and pitch. We describe the energy densities for each of the solutions in terms of the elastic properties and geometric shape of the unstrained rod, which allows determining the selection mechanism for the preferred helical configurations. This analysis promises to be a fundamental tool for understanding the close connection between the study of elastic deformations in thin rods and coarse-grained models with widespread applications in the natural sciences

First Principle Study of Capping Energies and Electronic States in Stoichiometric and Nonstoichiometric PbSe Nanoclusters

A large variety of PbSe nanoclusters have been modeled at the DFT level, to study their structure, their affinity for different ligands and their electronic properties, also depending on surface passivation. The clusters are extracted from the bulk rock salt structure with cubic, prism, truncated cubic, cuboctahedral and octahedral shape and they are fully relaxed, before computing the addition energies of methylamine and formate anions in different positions, to model the process of surface passivation. Then the density of states of all the clusters is computed, to study in particular the band gap and the behavior of the so-called intragap states, which affect the photophysical properties of the nanoparticles, also acting as trap states for charge carriers. We confirm the strong relationship between nanocluster off-stoichiometry and intragap states: such states can be localized on the surface, in the bulk or delocalized over the nanoparticle, according to the source of off-stoichiometry. The ability of different ligands to eliminate the intragap states are tested and discussed, also proposing nonstandard capping molecules